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Root Systems & Clifford Algebras Dechant, Pierre-Philippe ORCID: https://orcid.org/0000-0002-4694-4010 (2017) Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence. In: Pure Mathematics Colloquium, 13th January 2017, University of St Andrews, Scotland. (Unpublished) Downloaded from: http://ray.yorksj.ac.uk/id/eprint/4008/ Research at York St John (RaY) is an institutional repository. It supports the principles of open access by making the research outputs of the University available in digital form. Copyright of the items stored in RaY reside with the authors and/or other copyright owners. Users may access full text items free of charge, and may download a copy for private study or non-commercial research. For further reuse terms, see licence terms governing individual outputs. Institutional Repository Policy Statement RaY Research at the University of York St John For more information please contact RaY at [email protected] Viruses, root systems and affine extensions Clifford algebras and exceptional root systems The Coxeter plane Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Pierre-Philippe Dechant Departments of Mathematics and Biology, York Centre for Complex Systems Analysis, University of York St Andrews Pure Maths Colloquium { January 13, 2017 Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Clifford algebras and exceptional root systems The Coxeter plane Main results New affine symmetry principle for viruses and fullerenes H3 (icosahedral symmetry) induces the E8 root system Each 3D root system induces a 4D root system This correspondence extends to exponents in the Coxeter plane (not just the original Trinity) and ADE Lie algebras Clifford algebra is a very natural framework for root systems and reflection groups in general Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Overview 1 Viruses, root systems and affine extensions Viruses Root systems Affine extensions 2 Clifford algebras and exceptional root systems Clifford basics E8 from the icosahedron 3D to 4D spinor induction Trinities and McKay correspondence 3 The Coxeter plane Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions The Icosahedron Rotational icosahedral group is I = A5 of order 60 Full icosahedral group is the Coxeter group H3 of order 120 (including reflections/inversion); generated by the root system icosidodecahedron Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Icosahedral viruses Two viral surface layouts: a T = 4 Caspar-Klug quasiequivalent triangulation, and a pseudo T = 7 defying Caspar-Klug theory, which is based on a kite-rhombus tiling instead (HPV) Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Root systems Root system Φ: set of a2 a1 + a2 vectors a in a vector space with an inner product such that 1. Φ \ a = {−a;ag 8 a 2 Φ −a1 a1 R 2. sa Φ = Φ 8 a 2 Φ Simple roots: express −(a1 + a2) −a2 every element of Φ via a Z-linear combination. (vja) reflection/Coxeter groups s : v ! s (v) = v − 2 a a a (aja) Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Cartan Matrices (ai jaj ) jaj j Cartan matrix of ai s is Aij = 2 = 2 cosqij (ai jai ) jai j 2 −1 A : A = 2 −1 2 Coxeter-Dynkin diagrams: node = simple root, no link = roots p p orthogonal, simple link = roots at 3 , link with label m = angle m . 4 5 n A3 B3 H3 I2(n) Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Lie groups to Lie algebras to Coxeter groups to root systems Lie group: manifold of continuous symmetries (gauge theories, spacetime) Lie algebra: infinitesimal version near the identity Non-trivial part is given by a root lattice Weyl group is a crystallographic Coxeter group: An;Bn=Cn;Dn;G2;F4;E6;E7;E8 generated by a root system. So via this route root systems are always crystallographic. Neglect non-crystallographic root systems I2(n);H3;H4 . Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions An affine Coxeter group is the extension of a Coxeter group by an aff affine reflection in a hyperplane not containing the origin sa0 whose geometric action is given by aff 2(a0jv) sa0 v = a0 + v − a0 (a0ja0) Non-distance preserving: includes the translation generator aff Tv = v + a0 = sa0 sa0 v Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions { A2 Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions { A2 Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions { A2 Affine extensions of crystallographic Coxeter groups lead to a tessellation of the plane and a lattice. Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Non-crystallographic Coxeter groups H2 ⊂ H3 ⊂ H4 t(a1 + a2) a1 + ta2 ta + a a2 1 2 −a1 a1 −(ta1 + a2) −a2 −(a1 + ta2) −t(a1 + a2) H2 ⊂ H3 ⊂ H4: 10, 120, 14,400 elements, the only Coxeter groups that generate rotational symmetries of order 5 linear combinations now in the extended integer ring 1 p p [t] = fa + tbja;b 2 g golden ratio t = (1 + 5) = 2cos Z Z 2 5 1 p 2p x2 = x + 1 t0 = s = (1 − 5) = 2cos t + s = 1;ts = −1 2 5 Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence G T A random translation would give 5 secondary pentagons, i.e. 25 points. Here we have degeneracies due to `coinciding points'. Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions of non-crystallographic root systems? Unit translation along a vertex of a unit pentagon G Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence G A random translation would give 5 secondary pentagons, i.e. 25 points. Here we have degeneracies due to `coinciding points'. Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions of non-crystallographic root systems? Unit translation along a vertex of a unit pentagon G T Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions of non-crystallographic root systems? Unit translation along a vertex of a unit pentagon G G T A random translation would give 5 secondary pentagons, i.e. 25 points. Here we have degeneracies due to `coinciding points'. Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions of non-crystallographic root systems? p 1 Translation of length t = 2 (1 + 5) ≈ 1:618 (golden ratio) G T Cartoon version of a virus or carbon onion. Would there be an evolutionary benefit to have more than just compact symmetry? The problem has an intrinsic length scale. Pierre-Philippe Dechant Root systems & Clifford algebras: from symmetries of viruses to E8 & an ADE correspondence Viruses, root systems and affine extensions Viruses Clifford algebras and exceptional root systems Root systems The Coxeter plane Affine extensions Affine extensions of non-crystallographic Coxeter groups 2D and 3D point arrays for applications to viruses, fullerenes, quasicrystals, proteins etc Two complementary ways to construct these ISSN 2053-2733 Acta Crystallographica Section A September 2013 Volume
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